In establishing the uniform consistency results, we use the fact that under the given assumptions, the maximum size of the terminal node becomes arbitrarily small in probability uniformly, as argued in [1]}. However, the uniform rate of convergence can be very low. As a random-split tree grows deeper and has an increasingly large number of terminal nodes, there exists, with a not-very-small probability, at least one terminal node that is created without enough splits on a certain covariate. To enhance the rate, splitting can potentially be done in a stratified fashion, by assigning a certain minimum number of splits to each variable. The effect of the tree building rules on the rate of convergence would be an interesting future study especially in the presence of noise variables in high-dimensional data settings.
